Hopf cyclic cohomology has emerged from the attempt
to unravel the local formula [1]} for the cyclic cohomological Chern character  [2]}
of the hypoelliptic spectral triple [3]} modeling the space of leaves of a foliation.
A close inspection of that formula brought to light a Hopf algebra of `moving frames'
\({\mathcal {H}}(n, {\mathbb {R}})\)  (abbreviated as \({\mathcal {H}}_n\) ), which appeared to assume
in the transverse geometry of foliations a role similar to that of
\(\mathop {\rm GL}\nolimits (n, {\mathbb {R}})\)  as structure group of the frame bundle of a manifold.
Further examination revealed that the the index formula in question
was in fact the expression of a characteristic
cocycle in the range of a certain cohomology specific to Hopf algebras.
In the particular case of \({\mathcal {H}}_n\) , this Hopf cyclic cohomology
turned out to be isomorphic to the Gelfand-Fuchs cohomology of the
Lie algebra \(\mathfrak {a}_n\)  of formal vector fields on \({\mathbb {R}}^n\)  which, together with its
relative versions, is well-known to encode the geometric characteristic classes of foliations.
The benefit of having the same classes recaptured
in Hopf cyclic cohomology is that the latter affords a direct characteristic map to the cyclic
cohomology of étale foliations groupoids.
Ultimately, the main outcome of the
aforementioned investigation was the realization that the Hopf cyclic cohomology
\(HP^\bullet ({\mathcal {H}}_n; {\mathbb {C}}_\delta )\)  of \({\mathcal {H}}(n, {\mathbb {R}})\)  together with its relative versions
\(HP^\bullet ({\mathcal {H}}_n, {\rm O}_n; {\mathbb {C}}_\delta )\)  and \(HP^\bullet ({\mathcal {H}}_n, \mathop {\rm GL}\nolimits (n, {\mathbb {R}}); {\mathbb {C}}_\delta )\)  provide
the appropriate repository for all geometric
characteristic classes of foliations in the framework of noncommutative geometry.
